Here's an outline of the standard proof of the Spectral Theorem
for self-adjoint oprators *T* on a finite-dimensional
*real* inner product space *V*. This is clearer
than Axler's approach (pp.134 ff.), and does not depend on
the Fundamental Theorem of Algebra. Since I don't know
how to get a ``lambda'' or ``epsilon'' in HTML (are there commands
for these?), I'll use *c* and *e* instead.
Our aim is to show that there is an orthonormal basis
*v*_{1}, *v*_{2}, ...,
*v*_{n} for *V* such that
*Tv*_{j} =
*c*_{j}v_{j}
for each *j*. It is enough to prove the existence
of a single eigenvector, i.e. of a nonzero vector
*v*_{1} such that
*Tv*_{1} =
*c*_{1}*v*_{1}
for some real number *c*_{1}. This is because
(using the fact that *T=T*^{*}) if *v*
is an eigenvector and <*v,v'*>=0 then <*Tv,v'*>=0.
Thus *T* may be regarded as a self-adjoint operator on
the orthogonal complement of *v*_{1}; if that
complement is not the zero space, it will then contain
an eigenvector *v*_{2} by the same argument,
which we then repeat for the orthogonal complement of the span of
*v*_{1} and *v*_{2}, and so on until
we obtain our basis. (This concluding part of the argument *is*
done by Axler (see 7.13); the difference is how we get
*v*_{1} in the first place.)

Assume that we have succeeded. Without loss of generality we may
list the eigenvalues *c*_{j} in decreasing order.
Then for all *v* in *V* we find that <*Tv,v*>
is at most *c*_{1} <*v,v*>, with equality
if and only if *Tv* = *c*_{1}*v*.
The ratio between <*Tv,v*> and <*v,v*> is called
the **Rayleigh quotient** *R*_{T}(v).
[Of course we must have *v* nonzero.] Note that
*R*_{T}(v) = *R*_{T}(av) for
any nonzero scalar *a*. This suggests the following strategy:
Find *v*_{1} which maximizes *R*_{T}(v)
subject to <*v,v*>=1, and show that this *v*_{1}
is an eigenvector.

The point is that *since V is finite-dimensional*,
the function from *V* to **R** taking
any *v* to <*Tv,v*> is continuous (it's just
a quadratic polynomial in the coordinates), and the unit
sphere {*v* : <*v,v*>=1} is compact (using
Heine-Borel and the continuity of the function taking
*v* to <*v,v*>). Now, on <*v,v*>=1,
the Rayleigh quotient simplifies to <*Tv,v*>.
So this function is bounded and attains its maximum.
Call that maximum *c*_{1}, and let
*v*_{1} be a vector that attains it, i.e.
a vector such that <*v*_{1},*v*_{1}>=1
<*Tv*_{1},*v*_{1}>=*c*_{1}.

It remains to show that
*Tv*_{1}=*c*_{1}*v*_{1}.
Let *u* be any vector in *V*. Then
<*T(v*_{1}*+eu),v*_{1}*+eu*>
is at most
*c*_{1}<*v*_{1}*+eu,v*_{1}*+eu*>
for any scalar *e*. The difference is a linear combination of
*e* and *e*^{2}; for it to be nonnegative
for all *e*, the *e* coefficient must vanish.
This coefficient is
2<*Tv*_{1}*-c*_{1}*v*_{1},*u*>.
Taking
*u=Tv*_{1}*-c*_{1}*v*_{1}
we find that
*Tv*_{1}*-c*_{1}*v*_{1}=0.
Therefore
*Tv*_{1}*=c*_{1}*v*_{1}
as desired.